Learning Objectives

By the end of this 90-minute lecture, you will be able to:

  1. Understand numbers as mathematical abstractions and variables as placeholders
  2. Recognize functions as mathematical abstractions of relationships between variables
  3. Define Composite function and inverse functions
  4. Recognize the relation between functions and equations and solve equations using the their graphs

R skills

  1. Define a function using vector operation.
  2. Plot functions
  3. Solve non-equations using graphs

Introduction: Mathematics as a languange for abstract objects

Mathematics is often called the “universal language” because it provides precise tools for describing patterns, relationships, and changes in our world. Today, we begin our journey into calculus by exploring its fundamental building blocks: numbers, variables, functions, and equations.

Think of this section as learning the alphabet before writing poetry. Just as poets use words to express complex emotions and ideas, mathematicians use functions to express complex relationships and changes.

Knowledge Point 1: Numbers as Mathematical Abstractions and Variables as Placeholders

The Power of Abstract Thinking

Abstract Numbers Concept
Abstract Numbers Concept

In the real world, we can observe two apples and three potatoes. However, we cannot observe “2” and “3” themselves. Yet everyone knows what “2” means and what “3” means, despite the fact that we never directly observe these abstract concepts.

This remarkable ability reveals something profound: everyone has the capacity to think abstractly! This is the fundamental capacity that makes mathematics possible. When you recognize that two apples and two bananas share the same “twoness,” you are already thinking like a mathematician. This abstraction—moving from concrete objects to abstract concepts—is the foundation of all mathematical reasoning.

Example 1.1: The Coffee Shop Story

Imagine you’re managing a coffee shop. Every day, you observe patterns: - The number of customers varies throughout the day - Your revenue depends on how many cups you sell

These observations involve numbers (like 50 customers, $300 revenue) and relationships between quantities. Mathematics gives us tools to work with these relations precisely.

Formal Presentation

Definition 1.1 (Numbers as Abstractions): Numbers are mathematical objects that represent quantities, measurements, or positions. They abstract away specific physical properties to focus on quantitative relationships.

Definition 1.2 (Variables): A variable is a symbol (usually a letter) that represents an unknown or changing quantity. Variables serve as placeholders that can take on different values.

Types of Numbers in Calculus:

  • Natural numbers (ℕ): 1, 2, 3, 4, … where \(1, 2, 3, 4 \in \mathbb{N}\)
  • Integers (ℤ): …, -2, -1, 0, 1, 2, … where \(-2, -1, 0, 1, 2 \in \mathbb{Z}\)
  • Rational numbers (ℚ): fractions like 1/2, -3/4 where \(\frac{1}{2}, -\frac{3}{4} \in \mathbb{Q}\)
  • Real numbers (ℝ): all numbers on the number line, including √2, π, e where \(\sqrt{2}, \pi, e \in \mathbb{R}\)
  • Complex numbers (\(\mathbb{C}\)): A complex number is a number that has both a real part and an imaginary part. It is written in the standard form:\(a+ib\), where \(a\) is the real part and \(ib\) is the imaginary part, and \(i\) is the imaginary unit with \(i^2=-1\).

Important Relationship: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\)

This means every natural number is also an integer, every integer is also a rational number, every rational number is also a real number, and every real number is also a complex number. In this unit we focus on real numbers.

Graphical Presentation of Real Numbers

Real number can be presented by a line with zoer in the middle. Any real number corresponds to one point on the line, and any point on the line corresponds to a real number.

# Create a number line visualization
x <- seq(-5, 5, by = 0.1)
y <- rep(0, length(x))

# Create the plot
plot(x, y, type = "l", lwd = 3, col = "black", 
     xlim = c(-5, 5), ylim = c(-0.5, 0.5),
     xlab = "Real Numbers", ylab = "", yaxt = "n",
     main = "The Real Number Line")

# Add tick marks and labels
points(c(-4, -3, -2, -1, 0, 1, 2, 3, 4), rep(0, 9), pch = 19, cex = 1.2)
text(c(-4, -3, -2, -1, 0, 1, 2, 3, 4), rep(-0.2, 9), 
     labels = c(-4, -3, -2, -1, 0, 1, 2, 3, 4))

# Highlight special numbers
points(pi, 0, pch = 19, col = "red", cex = 1.5)
text(pi, 0.3, "π ≈ 3.14", col = "red", cex = 0.8)

points(-sqrt(2), 0, pch = 19, col = "blue", cex = 1.5)
text(-sqrt(2), 0.3, "-√2 ≈ -1.41", col = "blue", cex = 0.8)

# Add arrows
arrows(-5.5, 0, -5, 0, length = 0.1, lwd = 2)
arrows(5, 0, 5.5, 0, length = 0.1, lwd = 2)
The Real Number Line showing different types of numbers

The Real Number Line showing different types of numbers

Variable as a placeholder for numbers

Key Learning Points:

  • Variables in Action: See how changing the input variables (customers, price) immediately affects the output (revenue)

  • Relationship: Revenue = customers × price demonstrates a constant relationship, regardless of how the values of the variables change.

Interactive Quiz 1.1

Knowledge Point 2: Functions as Mathematical Notions of Relations

Example 1.1 The Coffee Shop Story (Revisited)

Let’s return to our coffee shop example with a mathematical perspective. Suppose you’ve established a fixed price of $3.50 per cup (prices don’t change very often in real businesses, so we treat price as a parameter). Your daily revenue depends linearly on the number of cups sold:

Revenue Function: R(Q) = 3.50Q

Where: - Q = number of cups sold (independent variable) - R = total revenue in dollars (dependent variable)
- 3.50 = price per cup (parameter)

This is a linear function because revenue increases at a constant rate with each additional cup sold.

Example 1.2 The Delivery Service Story

Consider a delivery service with a tiered pricing structure: - Short distances (0-5 km): $10 base fee - Medium distances (5-15 km): $10 + $2 per additional km beyond 5 km - Long distances (15+ km): $10 + $20 + $1.50 per additional km beyond 15 km

Piecewise Function: \[C(d) = \begin{cases} 10 & \text{if } 0 \leq d \leq 5 \\ 10 + 2(d-5) & \text{if } 5 < d \leq 15 \\ 10 + 20 + 1.5(d-15) & \text{if } d > 15 \end{cases}\]

This is a piece-wise linear function with different slopes for different distance ranges.

Formal Definition

Definition 1.3 (Function): A function is a rule that assigns to each input value exactly one output value. We write f(x) = y, where x is the input (independent variable) and y is the output (dependent variable). \(f: \mathbb R \to \mathbb R\)

Key Properties:

  1. Domain: The set of all possible input values

  2. Range: The set of all possible output values

  3. One-to-one correspondence: Each input produces exactly one output

For a real function: \(f: \mathbb R \to \mathbb R\)

Function Presentation Functions can be represented in multiple equivalent ways:

  1. Tabular Representation: Discrete input-output pairs

  2. Graphical Representation: Visual plot of the relationship

  3. Algebraic Representation: Mathematical formula

  4. Computational Representation: Algorithm or code

Example 1.3 The Weather Station Story

A weather station records temperature throughout the day. The same data can be presented in multiple ways: - Table: Hour-by-hour temperature readings - Graph: A smooth curve showing temperature changes - Formula: A mathematical equation that predicts temperature - Code: An algorithm that computes temperature values

Each representation reveals different aspects of the relationship and serves different purposes. For example, a table gives exact values at specific times, a graph shows overall patterns and trends, a formula allows calculation of temperature at any time, and code enables efficient computation for large datasets.

Multiple Representations: The Weather Station Example

# Create a comprehensive example showing all representations
set.seed(123)

# Define a temperature function (simplified model)
temperature_func <- function(hour) {
  # Base temperature with daily variation
  base_temp <- 20
  daily_variation <- 8 * sin(2 * pi * (hour - 6) / 24)
  noise <- rnorm(length(hour), 0, 1)
  return(base_temp + daily_variation + noise)
}

# Generate data
hours <- 0:23
temps <- temperature_func(hours)

# Create a 2x2 layout
par(mfrow = c(2, 2), mar = c(4, 4, 3, 2))

# 1. Table representation (as a plot)
plot(1, type = "n", xlim = c(0, 10), ylim = c(0, 10), 
     xlab = "", ylab = "", main = "Tabular Representation", axes = FALSE)
text(5, 9, "Hour | Temperature (°C)", cex = 1.2, font = 2)
for(i in 1:8) {
  text(5, 8.5 - i*0.8, paste(hours[i], "|", round(temps[i], 1)), cex = 1)
}
text(5, 1, "... (continues for 24 hours)", cex = 0.9, style = 3)

# 2. Graphical representation
plot(hours, temps, type = "b", pch = 19, col = "blue", lwd = 2,
     main = "Graphical Representation", 
     xlab = "Hour of Day", ylab = "Temperature (°C)")
grid()

# 3. Smooth function overlay
hours_smooth <- seq(0, 23, by = 0.1)
temps_smooth <- 20 + 8 * sin(2 * pi * (hours_smooth - 6) / 24)
plot(hours_smooth, temps_smooth, type = "l", lwd = 3, col = "red",
     main = "Algebraic Representation", 
     xlab = "Hour of Day", ylab = "Temperature (°C)")
text(12, 25, "T(h) = 20 + 8sin(2π(h-6)/24)", cex = 1.1, font = 2)
grid()

# 4. Code representation
plot(1, type = "n", xlim = c(0, 10), ylim = c(0, 10), 
     xlab = "", ylab = "", main = "Computational Representation", axes = FALSE)
code_lines <- c(
  "temperature <- function(hour) {",
  "  base_temp <- 20",
  "  variation <- 8 * sin(2*pi*(hour-6)/24)",
  "  return(base_temp + variation)",
  "}"
)
for(i in 1:length(code_lines)) {
  text(0.5, 9 - i*1.2, code_lines[i], cex = 0.9, adj = 0, family = "mono")
}
Multiple Representations of the Same Function

Multiple Representations of the Same Function 

par(mfrow = c(1, 1))  # Reset layout

Common Function Types

We will focus on presentation of real functions as mathematical formulas, computer codes, and graphs.

\[y = f(x) = 0.5x+2\] Using R code we can write “\(y=0.5*x+2\)”. This is one way to define a function for each input value in \(x\) we have a unique corresponding output value of \(y\). The method is called vectorized operation because \(x\) can be a vector i.e. a sequence of different values. Then \(y\) will be also a correponding vector of the same length.

To use graph to present the function we need specify the domain over which we want to plot the function. A sample code of plotting the function \(y = 0.5x+2\) is here.

x<-seq(-4,8,0.01)
y<- 0.5*x + 2
plot(x,y,type="l")

You can use the following R environment to test the above R code.

Examples of often used functions

  • Linear function \(y = a + bx\)
    • with \(a = 2\) and \(b=1\).
  • Quadratic function \(y = ax^2+bx+c\)
    • with \(a=1\), \(c=-2\).
  • Polynomial function \(y = a_nx^n+a_{n-1}x^{n-1}+...+a_1x + a_0\),
    • with \(n=3\), \(a_3=0.3\), \(a_2=0\),\(a_1=-1\), and \(a_0=1\).
  • Exponential function \(ab^x\)
    • with \(a=1\), \(b=2\)
  • Logarithmic function \(y = \log_b(x)\),
    • with \(b=e\).

Exercises

Please plot the above functions one by one over the domain of \((-4,8)\) using the R environment above or using your own local RStudio.

Graphical Presentation: Comparing All Function Types

The following graph shows all six common function types over the interval [-4, 8], demonstrating how different functions describe different relationships between dependent and independent variables:

# Create a comprehensive comparison of all function types
x <- seq(-4, 8, by = 0.1)

# Define all function types
linear_func <- 0.5 * x + 2
quadratic_func <- 0.2 * (x - 2)^2 + 1
polynomial_func <- 0.02 * x^3 - 0.1 * x^2 + x + 3
exponential_func <- 2^(0.5 * x)
logarithmic_func <- ifelse(x > 0, 2 * log(x) + 5, NA)
trigonometric_func <- 3 * sin(x) + 5

# Create the plot
plot(x, linear_func, type = "l", lwd = 3, col = "#007bff", 
     ylim = c(-5, 15), xlim = c(-4, 8),
     main = "Comparison of Function Types: Different Relations Between Variables",
     xlab = "Independent Variable (x)", ylab = "Dependent Variable f(x)",
     cex.main = 1.2, cex.lab = 1.1)

# Add all other functions
lines(x, quadratic_func, lwd = 3, col = "#6f42c1")
lines(x, polynomial_func, lwd = 3, col = "#fd7e14")
lines(x, exponential_func, lwd = 3, col = "#dc3545")
lines(x[x > 0], logarithmic_func[x > 0], lwd = 3, col = "#20c997")
lines(x, trigonometric_func, lwd = 3, col = "#6c757d")

# Add grid for better readability
grid(col = "lightgray", lty = 2)

# Add comprehensive legend
legend("topleft", 
       legend = c("Linear: f(x) = 0.5x + 2", 
                  "Quadratic: f(x) = 0.2(x-2)² + 1",
                  "Polynomial: f(x) = 0.02x³ - 0.1x² + x + 3",
                  "Exponential: f(x) = 2^(0.5x)",
                  "Logarithmic: f(x) = 2ln(x) + 5",
                  "Trigonometric: f(x) = 3sin(x) + 5"),
       col = c("#007bff", "#6f42c1", "#fd7e14", "#dc3545", "#20c997", "#6c757d"),
       lwd = 3, cex = 0.9, bg = "white")

# Add annotations to highlight key differences
text(-2, 12, "Different functions capture\ndifferent types of relationships:", 
     cex = 1.1, font = 2, adj = 0)
text(-2, 10, "• Linear: Constant rate of change", cex = 0.9, adj = 0)
text(-2, 9.2, "• Quadratic: Accelerating change", cex = 0.9, adj = 0)
text(-2, 8.4, "• Exponential: Explosive growth", cex = 0.9, adj = 0)
text(-2, 7.6, "• Logarithmic: Diminishing growth", cex = 0.9, adj = 0)
text(-2, 6.8, "• Trigonometric: Periodic patterns", cex = 0.9, adj = 0)
text(-2, 6.0, "• Polynomial: Complex curves", cex = 0.9, adj = 0)
Comparison of Common Function Types over [-4, 8]

Comparison of Common Function Types over [-4, 8]

We have so many different types of functions that respond to input variable in different ways over different intervals because in the real word the relation bwtween different variables are so different.

Interactive Quiz 1.2

Knowledge Point 3: Composite Function and Inverse Function: Exponential and Logarithmic Functions

Example 1.4: The Startup Growth Story

TechStart Inc. is analyzing its growth patterns. The company’s user base grows exponentially: \(U(t) = 1000 \cdot e^{0.2t}\) where \(t\) is time in months.

The marketing team asks: “If we want to reach 5000 users, how many months will it take?”

This question requires finding the inverse of the exponential function, that is, from the value of the output variable we want to know what is the value of the input variable.

Additionally, the company’s revenue depends on both user growth and pricing strategy: \(P(u) = 10u^{0.8}\). How does the company’s revenue depend on time? This leads to a composite function \(R(t) = P(U(t))\).

Definition 1.4: Composite Functions

Given functions \(f: A \to B\) and \(g: B \to C\), the composite function \((g \circ f): A \to C\) is defined by: \[(g \circ f)(x) = g(f(x))\]

Properties: - Associative: \((h \circ g) \circ f = h \circ (g \circ f)\) - Not commutative: \(g \circ f \neq f \circ g\) (in general)

In the example above, we have company’s growth dynamic \(U(t)=1000e^{0.2t)\) and price strategy \(P(u) = 10u^0.8\).Then the price dynamic is given by

\[P(t) = P(U(t))=10(1000e^{0.2t})^0.8=10^{3.4}e^{0.16t}\] In short a composite function is a function of function.

Definition 1.5: Inverse Functions

A function \(f: A \to B\) has an inverse function \(f^{-1}: B \to A\) if: \[f^{-1}(f(x)) = x \text{ for all } x \in A\] \[f(f^{-1}(y)) = y \text{ for all } y \in B\]

Existence condition: \(f\) must be one-to-one (injective) and onto (surjective).

Example of composite function: Exchange rate: 1 AUD = 0.6 USD, 1 USD = 150 YEN

\(Y_{USD} = f(X_{AUD}) = 0.6 X_{AUD}\),

\(Z_{YEN} = g(Y_{USD}) = 150 Y_{USD}\)

\(Z_{YEN} = (g\circ f )(X_{AUD}) = g(f(X_{AUD}))=150*(0.6*X_{AUD}) = 90X_{AUD}\)

Exponential and Logarithmic Functions

Exponential Function: \(f(x) = a^x\) where \(a > 0, a \neq 1\)

  • Natural exponential: \(e^x\) where \(e \approx 2.71828\)

  • Properties: \(a^{x+y} = a^x \cdot a^y\), \(a^{xy} = (a^x)^y\)

Logarithmic Function: \(g(x) = \log_a(x)\) where \(a > 0, a \neq 1\)

  • Natural logarithm: \(\ln(x) = \log_e(x)\)

  • Inverse relationship: \(a^{\log_a(x)} = x\) and \(\log_a(a^x) = x\)

A function \(y = f(x)\) and its inverse function \(y = f^{-1}(x)\) are symmetric about the 45 degree line.

f <- function(x) exp(x)
f_inv <- function(x) log(x)

# Generate x values
x_vals <- seq(-2, 2, by = 0.1)
y_vals <- f(x_vals)
x_inv_vals <- seq(0.1, exp(2), by = 0.1)
y_inv_vals <- f_inv(x_inv_vals)

# Set up the plot
plot(x_vals, y_vals, type = "l", col = "blue", lwd = 2,
     xlim = c(-2, 5), ylim = c(-2, 8),
     xlab = "x", ylab = "y",
     main = "Function and Its Inverse: Symmetry About y = x")

# Plot the inverse
lines(x_inv_vals, y_inv_vals, col = "red", lwd = 2)

# Plot the 45-degree line y = x
abline(a = 0, b = 1, col = "gray", lty = 2, lwd = 2)

For $f(x) = x^2 +2 $ and \(g(y) = e^y\) we have \((g\circ f)(x) = g(f(x))=e^{x^2+2}\). In R we can define the composite function in a similar way.

x <- seq(0, 2,0.1)
y <- x^2 + 3

z<- exp(x^2+3)

plot(x, z, type = "l", col = "blue", lwd = 2, 
     main = "Composite Function",
     xlab = "x values", ylab = "exp(x² + 3)")

Business Applications

Exponential Growth Models:

  • Population growth: \(P(t) = P_0 e^{rt}\)

Why is the exponential good for describing growth process? Let say you invest $1000 ant it grows \(r=5\%\) per year. After \(t\) years

\[Amount = 1000\times(1+r)^t=1000\times(1+r)^t\] This is exponential growth because time \(t\) is in the exponent! Now imagine you don’t just grow once per year, but every second, or continuously. How is the annual growth linked to the continuous growth? This special case is taken into account by changing the base from \((1+r)\) to \(e=2.718\)

The general exponential growth formula becomes:

\[Amount = 1000\times e^{rt}= 1000\times e^{0.05t}\]

t <- seq(1,5,0.1)

A_a = 1000*1.05^t
A_e = 1000*exp(0.05*t)

plot(t,A_a,type = "l")
lines(t,A_e,col="red")

  • Compound interest: \(A(t) = P(1 + r)^t\),

  • Technology adoption: \(N(t) = L/(1 + e^{-k(t-t_0)})\) (logistic)

In early stages, a new technology (like smartphones, electric cars, or social media platforms) can spread quickly:

  • At first: few users, but growth accelerates.

  • But eventually: most people who want it already have it.

  • So growth slows, and adoption levels off.

The logistic function models S-shaped (sigmoid) growth:

\[Adoption(t) = \frac{L}{1+e^{-k(t-t_0)}}\] Where:

  • \(L\) = maximum possible adoption (often called the carrying capacity)

  • \(k\) = how fast adoption spreads

  • \(t_0\) = the “midpoint” (when adoption hits 50%)

t <-seq(0,10,0.1)

Adoption <- 30/(1+exp(-1*(t-3)))

plot(t,Adoption,type="l")

Logarithmic Applications:

  • Time to reach target: \(t = \frac{\ln(N/N_0)}{r}\)

  • Elasticity of demand: \(\epsilon = \frac{d \ln Q}{d \ln P}\)

  • Information theory: \(H = -\sum p_i \ln(p_i)\)

Sumation Notation as a specific function

Calculate the following without using computer

  1. Calculate without using computer the sum of \(\sum_{i=1}^{10} i\) and Write R code to complete this task.

\[\sum_{i=1}^{10} i = 1+2+3+4+5+6+7+8+9+10 = 1+9 + 2+8 + 3+7 + 4+6 +5 = 55\]

# a)
y = 0
for (i in 1:10) y = y + i
y
## [1] 55
  1. Calculate without using computer the sum of \(\sum_{i=0}^n r^i\) and write R code to complete this task. \[S_n = \sum_{i=0}^n r^i\] \[rS_n = \sum_{i=1}^{n+1} r^i\] \[S_n-rS_n = \sum_{i=0}^{n} r^i - \sum_{i=1}^{n+1} r^i = r^0-r^{n+1}=S_n(1-r)\] \[S_n = \frac{r^0-r^{n+1}}{1-r}\]
#b)
r = 0.6
y=0
for (i in 0:10) y = y + r^i
y
## [1] 2.49093

Experience the power of R in computing complex summation expressions by modifying and running the R code above in the following R environment.

Interactive Quiz 1.3

Knowledge Point 4: Functions and Equations - Solving for Roots

Verbal Scenario: The Business Analyst

A business analyst encounters various equation-solving challenges:

Example 1.5 The Break-Even Analysis

A startup company wants to find its break-even point where total revenue equals total cost:

  • Revenue Function: R(Q) = 25Q (selling price $25 per unit)
  • Cost Function: C(Q) = 5000 + 15Q (fixed costs $5000, variable cost $15 per unit)

To find break-even: Set R(Q) = C(Q) 25Q = 5000 + 15Q 10Q = 5000 Q = 500 units

4.2 The Market Equilibrium

Finding where supply equals demand:

  • Supply Function: Qs = -100 + 2P (suppliers willing to sell)
  • Demand Function: Qd = 300 - P (consumers willing to buy)

At equilibrium: Qs = Qd -100 + 2P = 300 - P 3P = 400 P = $133.33, Q = 166.67 units

These are examples of solving equations to find where functions intersect.

Formal Presentation

Definition 1.6 (Equation): An equation is a mathematical statement that two expressions are equal. When one side is zero, we call the solutions roots or zeros.

Definition 1.7 (Root/Zero): A root of f(x) = 0 is a value x = r such that f(r) = 0.

Solving an equation problem can always be reformulated as the problem of finding the roots by moving all expression to the left-hand side.

Types of Equations and Solution Methods:

1. Linear Equations: ax + b = 0

  • Solution: x = -b/a
  • Always has exactly one solution (if a ≠ 0)

Concrete Example: 3x - 9 = 0 - Solution: x = 9/3 = 3 - Graphical interpretation: The line y = 3x - 9 crosses the x-axis (y = 0) at x = 3

2. Quadratic Equations: ax² + bx + c = 0

  • Quadratic formula: x = (-b ± √(b² - 4ac))/(2a)
  • Can have 0, 1, or 2 real solutions

Adjust the values of a, b, and c in the interactive graph below to investigate the relationship between the coefficients and the roots of the equation.

  • Two real solutions: The blue curve intersects the x-axis at two distinct points (two real roots).

  • One real solution: The curve touches the x-axis at exactly one point (a repeated real root).

  • No real solutions: The curve lies entirely above or below the x-axis (no real roots).

Quadratic Equation Summary Table

Case Discriminant (b² - 4ac) Nature of Roots Root Formulas
Two distinct real roots b² - 4ac > 0 Real and unequal \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
One real root (double) b² - 4ac = 0 Real and equal (repeated) x = -b/(2a)
No real roots (complex) b² - 4ac < 0 Complex conjugates x = (-b ± i√|b² - 4ac|)/(2a)

3. Higher-order Polynomial Equations: aₙxⁿ + … + a₁x + a₀ = 0

  • May require numerical methods for n > 4
  • Can have up to n real roots

Concrete Example: Fourth-order polynomial with 4 roots - Factored form: y = (x + 2)(x - 1)(x - 3)(x - 5) - Expanded: \(y = x^4 -7x^3+5x^2+31x-30\) - Four roots: x = -2, 1, 3, 5

4. Transcendental Equations: Involving exponential, logarithmic, or trigonometric functions

  • Usually require numerical methods

Concrete Example: eˣ - 3 = 0 - Solution: x = ln(3) ≈ 1.099 - Graphical interpretation: The curve y = eˣ - 3 crosses the x-axis at x = ln(3)

Solving an equation

You can input an equation in the following input box and click the button “Update Function”. Then the intersections between the function curve and the horizontal axis are the roots of the function and hence the solutions of the equation. You can use mouse the move the blue point to find the value of the respective solutions.

Interactive Quiz 1.4

Comprehensive Review Questions

This section provides review and application questions designed to assess understanding at five different levels. Each level builds upon the previous one, fostering deeper comprehension and the ability to apply concepts in increasingly complex situations.

https://pchen-exercisesfeedback.hf.space

End of Section 1 Lecture Notes

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